A tension roller for drive belts is disclosed, for example, in DE-GM8120032. The support body for the tension roller disclosed in this reference consists of two cup-shaped sleeves one of which is inserted in the other. A conventional rolling bearing is clamped between the side end face of one of the sleeves and a bulge in the region of the outer surface of the other sleeve, a roller being arranged on the outer ring of the rolling bearing. Eccentrically positioned bores are provided in the sleeves, and a screw extends therethrough for affixing the tension roller to a machine frame, whereby the tension roller as a whole is eccentrically pivotal about the screw. The screw extends through the inner space of the sleeve, so that the wall thickness of the sleeve limits the maximum eccentric position of the bore for the screw. Especially in the event that smaller roller bearings are to be provided for loading considerations, the tension range in known arrangements is ofttimes not achieved due to the small bores of the rolling bearings and the limited eccentric arrangements of the bores for the screws connected thereto. This disadvantage restricts the range of variation between rolling bearing size, tension roller diameter, and tension range in the known tension roller arrangement, so that the optimum tension roller arrangement has not been achieved.